| Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclscls00 | Structured version Visualization version GIF version | ||
| Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.) |
| Ref | Expression |
|---|---|
| ntrcls.o | ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) |
| ntrcls.d | ⊢ 𝐷 = (𝑂‘𝐵) |
| ntrcls.r | ⊢ (𝜑 → 𝐼𝐷𝐾) |
| Ref | Expression |
|---|---|
| ntrclscls00 | ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ntrcls.o | . . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
| 2 | ntrcls.d | . . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵) | |
| 3 | ntrcls.r | . . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
| 4 | 1, 2, 3 | ntrclsfv1 44068 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾) |
| 5 | 4 | fveq1d 6908 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐾‘∅)) |
| 6 | 2, 3 | ntrclsbex 44047 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
| 7 | 1, 2, 3 | ntrclsiex 44066 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
| 8 | eqid 2737 | . . . . 5 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼) | |
| 9 | 0elpw 5356 | . . . . . 6 ⊢ ∅ ∈ 𝒫 𝐵 | |
| 10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∅ ∈ 𝒫 𝐵) |
| 11 | eqid 2737 | . . . . 5 ⊢ ((𝐷‘𝐼)‘∅) = ((𝐷‘𝐼)‘∅) | |
| 12 | 1, 2, 6, 7, 8, 10, 11 | dssmapfv3d 44032 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅)))) |
| 13 | 5, 12 | eqtr3d 2779 | . . 3 ⊢ (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅)))) |
| 14 | dif0 4378 | . . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵 | |
| 15 | 14 | fveq2i 6909 | . . . . . 6 ⊢ (𝐼‘(𝐵 ∖ ∅)) = (𝐼‘𝐵) |
| 16 | id 22 | . . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘𝐵) = 𝐵) | |
| 17 | 15, 16 | eqtrid 2789 | . . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵) |
| 18 | 17 | difeq2d 4126 | . . . 4 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵 ∖ 𝐵)) |
| 19 | difid 4376 | . . . 4 ⊢ (𝐵 ∖ 𝐵) = ∅ | |
| 20 | 18, 19 | eqtrdi 2793 | . . 3 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅) |
| 21 | 13, 20 | sylan9eq 2797 | . 2 ⊢ ((𝜑 ∧ (𝐼‘𝐵) = 𝐵) → (𝐾‘∅) = ∅) |
| 22 | pwidg 4620 | . . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵) | |
| 23 | 6, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵) |
| 24 | 1, 2, 3, 23 | ntrclsfv 44072 | . . 3 ⊢ (𝜑 → (𝐼‘𝐵) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵)))) |
| 25 | 19 | fveq2i 6909 | . . . . . 6 ⊢ (𝐾‘(𝐵 ∖ 𝐵)) = (𝐾‘∅) |
| 26 | id 22 | . . . . . 6 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅) | |
| 27 | 25, 26 | eqtrid 2789 | . . . . 5 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘(𝐵 ∖ 𝐵)) = ∅) |
| 28 | 27 | difeq2d 4126 | . . . 4 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = (𝐵 ∖ ∅)) |
| 29 | 28, 14 | eqtrdi 2793 | . . 3 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = 𝐵) |
| 30 | 24, 29 | sylan9eq 2797 | . 2 ⊢ ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼‘𝐵) = 𝐵) |
| 31 | 21, 30 | impbida 801 | 1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |